Extracting quantitative dielectric properties from pump-probe spectroscopy

Optical pump-probe spectroscopy is a powerful tool for the study of non-equilibrium electronic dynamics and finds wide applications across a range of fields, from physics and chemistry to material science and biology. However, a shortcoming of conventional pump-probe spectroscopy is that photoinduced changes in transmission, reflection and scattering can simultaneously contribute to the measured differential spectra, leading to ambiguities in assigning the origin of spectral signatures and ruling out quantitative interpretation of the spectra. Ideally, these methods would measure the underlying dielectric function (or the complex refractive index) which would then directly provide quantitative information on the transient excited state dynamics free of these ambiguities. Here we present and test a model independent route to transform differential transmission or reflection spectra, measured via conventional optical pump-probe spectroscopy, to changes in the quantitative transient dielectric function. We benchmark this method against changes in the real refractive index measured using time-resolved Frequency Domain Interferometry in prototypical inorganic and organic semiconductor films. Our methodology can be applied to existing and future pump-probe data sets, allowing for an unambiguous and quantitative characterisation of the transient photoexcited spectra of materials. This in turn will accelerate the adoption of pump-probe spectroscopy as a facile and robust materials characterisation and screening tool.


SI 1 -Retrieving the Stimulated Emission Band from an Optical Constant Based Analysis
The KK analysis presented does not include the stimulated emission (SE) band. This is because the approach focuses solely on constructing the photoexcited weighed density of states in the material. Just like photoluminescence, which is also not included in this model, changes in the net transmission that result from radiation from the material rather than changes in optical constants are not explicitly captured in our analysis. In order to fully recover the stimulated emission band from this material, there exist two options depending on the magnitude of the Stokes shift: a) If the Stokes shift is large, one can study the photobleaching-like negative spectral signatures in ∆ # that appear shifted from the expected ground state photobleaching feature (based on the static # ), just as is typically done in transient absorption spectroscopy. b) However, if the Stokes shift is small, one can combine the methodology we provide to calculate ∆ #_&&' that includes the SE band with the Frequency Domain Interferometry (FDI) retrieved ∆ #_()* which is sensitive solely to the real part of the refractive index and therefore does not pick up the SE band. The spectrum of ∆ #_&&' -∆ #_()* should contain purely the SE band.

SI 2 -Fresnel's Equations for Reflection and Transmission of a Thin Film and Wafer
The relationship between the spectral properties and the dielectric function/refractive index of the material are given by the well-known Fresnel Equations by solving Maxwell's equations with boundary conditions at the interfaces. Ignoring Fabry-Pérot resonances the thin film reflection and transmission are given by, r and t being the electric field complex reflection and transmission amplitudes respectively. Similar expressions including Fabry-Pérot resonances and for 2D materials can also be derived 1 . For the sake of brevity we use the definitions above for our thin film samples studied in reflection. For a GaAs wafer the expression for the top surface reflection, is the only one that is needed as there is no transmission through the material.
We note that while we have ignored the interference based Fabry-Pérot resonances, similar expression for the transmission and reflection for a multilayer sample can be calculated in both the coherent and incoherent regimes using the transfer matrix method. 2

SI 3 -Variational KK Fitting of CsPbBr3 Data
The ellipsometric and transmission data from 3,4 were taken and fit first to 3 Lorentz oscillators. These were fixed and the residues were variationally fit the triangular oscillators reported in 5 . The final fits are shown in Fig S3 a), b) and c). These fits were used to calculate the derivatives in Eqn (4) of the main text. The differential transmission at a given time delay was then fit using the differential dielectric function expansion and the final fit including the variational oscillators are shown in Fig S3 d). The last step was repeated for all time points.

SI 5 -Variational KK Fitting of Pentacene
The ellipsometric data from 6 and measured transmission were taken and fit first to 5 Lorentz oscillators. These were fixed and the residues were variationally fit to the triangular oscillators reported in 5 . The final fits are shown in Fig S5 a), b) and c). These fits were used to calculate the derivatives in Eqn (4) of the main text. The differential transmission at a given time delay was then fit using the differential dielectric function expansion and the final fit including the variational oscillators are shown in Fig S5 d). The last step was repeated for all time points.

SI 6 -Variational KK Fitting of GaAs
The ellipsometric data from 7 was taken and fit first to 5 Lorentz oscillators. These were fixed and the residues were variationally fit the triangular oscillators reported in 5 . The final fits are shown in Fig S6 a) and b). c) shows the retrieved reflection spectrum compared to that measured by 8 . These fits were used to calculate the derivatives in Eqn (4) of the main text except here we calculate the differential reflectance instead of the transmission. These are used to then fit the measured differential reflectance at a given time delay using the differential dielectric function expansion and the final fit including the variational oscillators are shown in Fig S6 d). The last step was repeated for all time points. Figure S6: KK Fitting of GaAs. a) and b) are the final variational fits of the static ellipsometry and c) is the recovered reflectivity data obtained from 6 compared to that from 7 . From the fits to this data the derivates are calculated and used to variationally fit the differential dielectric function to the transient reflection data as shown in d).